Mexican hat potential · order parameter dynamics · massless Goldstone modes
When a system with a continuous symmetry settles into a ground state that breaks that symmetry, massless Goldstone bosons appear — one for each broken symmetry generator (Goldstone's theorem, 1962). The paradigm example is a complex scalar field φ = ρeiθ in a Mexican hat potential V(φ) = λ(|φ|² − v²)². The radial mode (ρ) is massive; the angular mode (θ) is massless.
The order parameter evolves under: φ̈ = −∂V/∂φ* − γφ̇
In the Standard Model, the Higgs mechanism gives mass to the Goldstone bosons by coupling them to gauge fields, turning them into longitudinal polarizations of the W±, Z bosons.
Massive mode: m² = 2λv² | Goldstone: m = 0